Average Error: 61.3 → 0.5
Time: 15.9s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\log \left(e^{\frac{\log 1 - \left(1 \cdot x + \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}}\right)\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\log \left(e^{\frac{\log 1 - \left(1 \cdot x + \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}}\right)
double f(double x) {
        double r62155 = 1.0;
        double r62156 = x;
        double r62157 = r62155 - r62156;
        double r62158 = log(r62157);
        double r62159 = r62155 + r62156;
        double r62160 = log(r62159);
        double r62161 = r62158 / r62160;
        return r62161;
}
↓
double f(double x) {
        double r62162 = 1.0;
        double r62163 = log(r62162);
        double r62164 = x;
        double r62165 = r62162 * r62164;
        double r62166 = 0.5;
        double r62167 = 2.0;
        double r62168 = pow(r62164, r62167);
        double r62169 = pow(r62162, r62167);
        double r62170 = r62168 / r62169;
        double r62171 = r62166 * r62170;
        double r62172 = r62165 + r62171;
        double r62173 = r62163 - r62172;
        double r62174 = r62165 + r62163;
        double r62175 = r62174 - r62171;
        double r62176 = r62173 / r62175;
        double r62177 = exp(r62176);
        double r62178 = log(r62177);
        return r62178;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.3
Target0.4
Herbie0.5
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + 0.4166666666666666851703837437526090070605 \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 61.3

    \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]

  2. Taylor expanded around 0 60.4

    \[\leadsto \frac{\log \left(1 - x\right)}{\color{blue}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}}\]

  3. Taylor expanded around 0 0.5

    \[\leadsto \frac{\color{blue}{\log 1 - \left(1 \cdot x + \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}\]
  4. Using strategy rm

  5. Applied add-log-exp0.5

    \[\leadsto \color{blue}{\log \left(e^{\frac{\log 1 - \left(1 \cdot x + \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}}\right)}\]

  6. Final simplification0.5

    \[\leadsto \log \left(e^{\frac{\log 1 - \left(1 \cdot x + \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}}\right)\]

Reproduce

herbie shell --seed 2019326 
(FPCore (x)
  :name "qlog (example 3.10)"
  :precision binary64
  :pre (and (< -1 x) (< x 1))

  :herbie-target
  (- (+ (+ (+ 1 x) (/ (* x x) 2)) (* 0.4166666666666667 (pow x 3))))

  (/ (log (- 1 x)) (log (+ 1 x))))